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<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">P</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>;<br/><b>let </b><font color="Maroon">p1</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>, <font color="Maroon">p2</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>;<br/>





<b>assume </b><a NAME="E1:16"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">P</font> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>
 ;<br/>

<b>then consider </b><font color="Maroon">f</font> being   <a href="struct_0.html#NM6">Function</a> of <a href="borsuk_1.html#K20">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P</font></span>)</span><b> such that </b><br/><a NAME="E3:16"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">f</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a>  &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> )
 <b>by </b><i><a class="ref" href="topreal1.html#D2">TOPREAL1:def 2</a></i>;<br/>
<b>set </b><font color="Maroon">Ex</font> =  <a href="treal_1.html#K3">L[01]</a> <span class="p1">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span>,<span class="p1">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span>;<br/>
<a NAME="E4:16"/><i><font color="Green">E33</font></i>: 
 <a href="treal_1.html#K3">L[01]</a> <span class="p1">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span>,<span class="p1">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a> 
 
<b>by </b><i><a class="ref" href="treal_1.html#T21">TREAL_1:21</a></i>;<br/>
<b>set </b><font color="Maroon">g</font> = <font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="treal_1.html#K3">L[01]</a> <span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span>,<span class="p2">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span></span>)</span>;<br/>
<a NAME="E5:16"/><i><font color="Green">E36</font></i>: 
<span class="p1">(<span class="default"><a href="treal_1.html#K3">L[01]</a> <span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span>,<span class="p2">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D17">BORSUK_1:def 17</a>, <a class="ref" href="treal_1.html#T8">TREAL_1:8</a>, <a class="ref" href="treal_1.html#T12">TREAL_1:12</a></i>;<br/>
<a NAME="E6:16"/><i><font color="Green">E37</font></i>: 
<span class="p1">(<span class="default"><a href="treal_1.html#K3">L[01]</a> <span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span>,<span class="p2">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D18">BORSUK_1:def 18</a>, <a class="ref" href="treal_1.html#T8">TREAL_1:8</a>, <a class="ref" href="treal_1.html#T12">TREAL_1:12</a></i>;<br/>
<a NAME="E7:16"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <a href="borsuk_1.html#K20">I[01]</a> 
 
<b>by </b><i>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E8:16"/><b>then </b><i><font color="Green">E38</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="treal_1.html#K3">L[01]</a> <span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span>,<span class="p2">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="topmetr.html#T27">TOPMETR:27</a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E9:16"/><b>then </b><i><font color="Green">E51</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="treal_1.html#K3">L[01]</a> <span class="p3">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span>,<span class="p3">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="treal_1.html#K3">L[01]</a> <span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span>,<span class="p2">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/>
<a NAME="E10:16"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="treal_1.html#K3">L[01]</a> <span class="p3">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span>,<span class="p3">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="relat_1.html#T47">RELAT_1:47</a></i>;<br/>
<a NAME="E11:16"/><b>then </b><i><font color="Green">E52</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="treal_1.html#K3">L[01]</a> <span class="p3">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span>,<span class="p3">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">P</font> = <font color="Maroon">P</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <b>by </b><i>, <a class="ref" href="topreal1.html#T4">TOPREAL1:4</a></i>;<br/>
<a NAME="E13:16"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="treal_1.html#K3">L[01]</a> <span class="p3">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span>,<span class="p3">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <a href="borsuk_1.html#K20">I[01]</a> 
 
<b>by </b><i>, , <a class="ref" href="topmetr.html#T27">TOPMETR:27</a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E14:16"/><b>then </b><i><font color="Green">E53</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="treal_1.html#K3">L[01]</a> <span class="p3">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span>,<span class="p3">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="borsuk_1.html#K20">I[01]</a>  &amp;  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="treal_1.html#K3">L[01]</a> <span class="p3">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span>,<span class="p3">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P</font></span>)</span> )
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">g</font> = <font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="treal_1.html#K3">L[01]</a> <span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span>,<span class="p2">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span></span>)</span> as   <a href="struct_0.html#NM6">Function</a> of <a href="borsuk_1.html#K20">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P</font></span>)</span> <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<a NAME="E16:16"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">g</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a> 
 
<b>by </b><i>, , <a class="ref" href="topmetr.html#T27">TOPMETR:27</a>, <a class="ref" href="tops_2.html#T71">TOPS_2:71</a></i>;<br/>
<a NAME="E17:16"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font>
 
<b>by </b><i>, , , <a class="ref" href="borsuk_1.html#D17">BORSUK_1:def 17</a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a>, <a class="ref" href="treal_1.html#T8">TREAL_1:8</a></i>;<br/>
<a NAME="E18:16"/>
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font>
 
<b>by </b><i>, , , <a class="ref" href="borsuk_1.html#D18">BORSUK_1:def 18</a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a>, <a class="ref" href="treal_1.html#T8">TREAL_1:8</a></i>;<br/>
<b>hence </b><a NAME="E19:16"/>
<font color="Maroon">P</font> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Maroon">p2</font>,<font color="Maroon">p1</font>
 <b>by </b><i>, , <a class="ref" href="topreal1.html#D2">TOPREAL1:def 2</a></i>;<br/>


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